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QB_EM_CALC_010
Question Bank • QB_EM_CALC_010
MCQ • 2 Marks
Section 1: Engineering Mathematics
Calculus
Subtopic: Total differentiation
Question Statement:
If the equation $F(x, y, z) = 0$ defines $z$ implicitly as a function of independent variables $x$ and $y$, then provided $\frac{\partial F}{\partial z} \ne 0$, the partial derivative $\frac{\partial z}{\partial x}$ is given by:
Options:
(A) $-\frac{\partial F/\partial x}{\partial F/\partial z}$
(B) $\frac{\partial F/\partial x}{\partial F/\partial z}$
(C) $-\frac{\partial F/\partial z}{\partial F/\partial x}$
(D) $\frac{\partial F/\partial z}{\partial F/\partial x}$
Step-by-Step Solution & Explanation
Taking the total differential of $F(x, y, z) = 0$:
$dF = \frac{\partial F}{\partial x} dx + \frac{\partial F}{\partial y} dy + \frac{\partial F}{\partial z} dz = 0$
To find $\frac{\partial z}{\partial x}$, hold $y$ constant ($dy = 0$):
$\frac{\partial F}{\partial x} dx + \frac{\partial F}{\partial z} dz = 0 \implies \frac{\partial z}{\partial x} = -\frac{\partial F/\partial x}{\partial F/\partial z}$
Hence, option A is correct.
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